Abstract

An optical heterodyne receiver is, in effect, both a receiver and an antenna. As an antenna it has an effective aperture or capture cross section AR(Ω) for plane wave signals arriving from any direction Ω. The wavefront alignment between signal and local-oscillator (LO) beams required for effective optical heterodyning may be summarized in the “antenna theorem” AR(Ω)dΩ=[η2¯/η¯2]λ2 where the moments of the quantum efficiency η are evaluated over the photosensitive surface. Thus, an optical heterodyne having effective aperture AR for signals arriving within a single main antenna lobe or field of view of solid angle ΩR is limited by the constraint ARΩR≈λ2. Optical elements placed in the signal and/or LO beam paths can vary the trade-off between AR and ΩR but cannot change their product.

It is also noted that an optical heterodyne is an insensitive detector for thermal radiation, since a thermal source filling the receiver’s field of view must have a temperature T≈[ln (1+ η¯)]−1hf/k to be detected with S/N≈1. Optical heterodyning can be useful in practical situations, however, for detecting Doppler shifts in coherent light scattered by liquids, gases, or small particles. Another antenna theorem applicable to this problem says that in a scattering experiment the received power will be ≲λ/4π times the transmitted power, where N is the density of scatterers and σ is the total scattering cross section of a single scatterer. The equality sign is obtained only when a single aperture serves as both transmitting and receiving aperture, or when two separate apertures are optimally focused at short range onto a common volume.

© 1966 Optical Society of America

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References

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  1. S. Jacobs, “The optical heterodyne,” Electronics, vol. 36(28), p. 29, July12, 1963.
  2. A. E. Siegman, S. E. Harris, B. J. McMurtry, “Optical heterodyning and optical demodulation at microwave frequencies,” in Optical Masers, J. Fox, Ed. New York: Polytechnic Press and Wiley, 1963, p. 511.
  3. B. M. Oliver, “Signal-to-noise ratios in photoelectric mixing,” Proc. IRE (Correspondence), vol. 49, pp. 1960–1961, December1961.
  4. H. A. Haus, C. H. Townes, B. M. Oliver, “Comments on ‘Noise in photoelectric mixing,’” Proc. IRE (Correspondence), vol. 50, pp. 1544–1545, June1962.
  5. S. Jacobs, P. Rabinowitz, “Optical heterodyning with a CW gaseous laser,” in Quantum Electronics III, P. Grivet, N. Bloembergen, Eds. New York: Columbia University Press1964, p. 481.
  6. P. Rabinowitz, J. LaTourrette, G. Gould, “AFC optical heterodyne detector,” Proc. IEEE (Correspondence), vol. 51, pp. 857–858, May1963.
    [Crossref]
  7. V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys., vol. 36, p. 1819, June1965.
    [Crossref]
  8. R. D. Kroger, “Motion sensing by optical heterodyne Doppler detection from diffuse surfaces,” Proc. IEEE (Correspondence)vol. 53, pp. 211–212, February1965.
    [Crossref]
  9. G. Gould et al.., “Coherent detection of light scattered from a diffusely reflecting surface,” Appl. Opt., vol. 3, p. 648, May1964.
    [Crossref]
  10. W. S. Read, D. L. Fried, “Optical heterodyning with noncritical angular alignment,” Proc. IEEE (Correspondence), vol. 51, p. 1787, December1963.
    [Crossref]
  11. W. S. Read, R. G. Turner, “Tracking heterodyne detection,” Appl. Opt., vol. 4, p. 1570, December1965.
    [Crossref]
  12. A. E. Siegman, “Detection and demodulation of laser beams,” presented at the 1964 International Symposium, IEEE Microwave Theory and Techniques Group, New York, N. Y.
  13. H. Z. Cummins, N. Knable, Y. Yeh, “Observation of diffusion broadening of Rayleigh scattered light,” Phys. Rev. Lett., vol. 12, p. 150, February10, 1964.
    [Crossref]
  14. Y. Yeh, H. Z. Cummins, “Localized fluid flow measurements with an He-Ne laser spectrometer,” Appl. Phys. Lett., vol. 4, p. 176May15, 1964.
    [Crossref]
  15. J. W. Foreman, E. W. George, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flow meter,” Appl. Phys. Lett., vol. 7, p. 77, August5, 1965.
    [Crossref]
  16. A. E. Siegman, “Spatially coherent detection of scattered radiation,” submitted for publication, IEEE Trans. on Antennas and Propagation.
  17. R. L. Smith, “Theory of photoelectric mixing at a metal surface,” Appl. Opt., vol. 3, p. 709, June1964.
    [Crossref]
  18. A. J. Bahr, “The effect of polarization selectivity on optical mixing in photoelectric surfaces,” Proc. IEEE (Correspondence), vol. 53, p. 513, May1965.
    [Crossref]
  19. This same point has been noted by R. E. Brooks, TRW Space Technology Labs, Redondo Beach, Calif., unpublished memorandum.
  20. Note added in proof:More recent calculations indicate that the limit seems to be given by Nσλ/4, i.e., a factor of πlarger than stated in the text. This “theorem” has yet to be proved in general, but an exact calculation for a single transmit-receive antenna with a Gaussian beam pattern leads to exactly Nσλ/4.

1965 (5)

V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys., vol. 36, p. 1819, June1965.
[Crossref]

R. D. Kroger, “Motion sensing by optical heterodyne Doppler detection from diffuse surfaces,” Proc. IEEE (Correspondence)vol. 53, pp. 211–212, February1965.
[Crossref]

W. S. Read, R. G. Turner, “Tracking heterodyne detection,” Appl. Opt., vol. 4, p. 1570, December1965.
[Crossref]

J. W. Foreman, E. W. George, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flow meter,” Appl. Phys. Lett., vol. 7, p. 77, August5, 1965.
[Crossref]

A. J. Bahr, “The effect of polarization selectivity on optical mixing in photoelectric surfaces,” Proc. IEEE (Correspondence), vol. 53, p. 513, May1965.
[Crossref]

1964 (4)

R. L. Smith, “Theory of photoelectric mixing at a metal surface,” Appl. Opt., vol. 3, p. 709, June1964.
[Crossref]

H. Z. Cummins, N. Knable, Y. Yeh, “Observation of diffusion broadening of Rayleigh scattered light,” Phys. Rev. Lett., vol. 12, p. 150, February10, 1964.
[Crossref]

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurements with an He-Ne laser spectrometer,” Appl. Phys. Lett., vol. 4, p. 176May15, 1964.
[Crossref]

G. Gould et al.., “Coherent detection of light scattered from a diffusely reflecting surface,” Appl. Opt., vol. 3, p. 648, May1964.
[Crossref]

1963 (3)

W. S. Read, D. L. Fried, “Optical heterodyning with noncritical angular alignment,” Proc. IEEE (Correspondence), vol. 51, p. 1787, December1963.
[Crossref]

S. Jacobs, “The optical heterodyne,” Electronics, vol. 36(28), p. 29, July12, 1963.

P. Rabinowitz, J. LaTourrette, G. Gould, “AFC optical heterodyne detector,” Proc. IEEE (Correspondence), vol. 51, pp. 857–858, May1963.
[Crossref]

1962 (1)

H. A. Haus, C. H. Townes, B. M. Oliver, “Comments on ‘Noise in photoelectric mixing,’” Proc. IRE (Correspondence), vol. 50, pp. 1544–1545, June1962.

1961 (1)

B. M. Oliver, “Signal-to-noise ratios in photoelectric mixing,” Proc. IRE (Correspondence), vol. 49, pp. 1960–1961, December1961.

Bahr, A. J.

A. J. Bahr, “The effect of polarization selectivity on optical mixing in photoelectric surfaces,” Proc. IEEE (Correspondence), vol. 53, p. 513, May1965.
[Crossref]

Brooks, R. E.

This same point has been noted by R. E. Brooks, TRW Space Technology Labs, Redondo Beach, Calif., unpublished memorandum.

Corcoran, V. J.

V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys., vol. 36, p. 1819, June1965.
[Crossref]

Cummins, H. Z.

H. Z. Cummins, N. Knable, Y. Yeh, “Observation of diffusion broadening of Rayleigh scattered light,” Phys. Rev. Lett., vol. 12, p. 150, February10, 1964.
[Crossref]

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurements with an He-Ne laser spectrometer,” Appl. Phys. Lett., vol. 4, p. 176May15, 1964.
[Crossref]

Foreman, J. W.

J. W. Foreman, E. W. George, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flow meter,” Appl. Phys. Lett., vol. 7, p. 77, August5, 1965.
[Crossref]

Fried, D. L.

W. S. Read, D. L. Fried, “Optical heterodyning with noncritical angular alignment,” Proc. IEEE (Correspondence), vol. 51, p. 1787, December1963.
[Crossref]

George, E. W.

J. W. Foreman, E. W. George, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flow meter,” Appl. Phys. Lett., vol. 7, p. 77, August5, 1965.
[Crossref]

Gould, G.

G. Gould et al.., “Coherent detection of light scattered from a diffusely reflecting surface,” Appl. Opt., vol. 3, p. 648, May1964.
[Crossref]

P. Rabinowitz, J. LaTourrette, G. Gould, “AFC optical heterodyne detector,” Proc. IEEE (Correspondence), vol. 51, pp. 857–858, May1963.
[Crossref]

Harris, S. E.

A. E. Siegman, S. E. Harris, B. J. McMurtry, “Optical heterodyning and optical demodulation at microwave frequencies,” in Optical Masers, J. Fox, Ed. New York: Polytechnic Press and Wiley, 1963, p. 511.

Haus, H. A.

H. A. Haus, C. H. Townes, B. M. Oliver, “Comments on ‘Noise in photoelectric mixing,’” Proc. IRE (Correspondence), vol. 50, pp. 1544–1545, June1962.

Jacobs, S.

S. Jacobs, “The optical heterodyne,” Electronics, vol. 36(28), p. 29, July12, 1963.

S. Jacobs, P. Rabinowitz, “Optical heterodyning with a CW gaseous laser,” in Quantum Electronics III, P. Grivet, N. Bloembergen, Eds. New York: Columbia University Press1964, p. 481.

Knable, N.

H. Z. Cummins, N. Knable, Y. Yeh, “Observation of diffusion broadening of Rayleigh scattered light,” Phys. Rev. Lett., vol. 12, p. 150, February10, 1964.
[Crossref]

Kroger, R. D.

R. D. Kroger, “Motion sensing by optical heterodyne Doppler detection from diffuse surfaces,” Proc. IEEE (Correspondence)vol. 53, pp. 211–212, February1965.
[Crossref]

LaTourrette, J.

P. Rabinowitz, J. LaTourrette, G. Gould, “AFC optical heterodyne detector,” Proc. IEEE (Correspondence), vol. 51, pp. 857–858, May1963.
[Crossref]

Lewis, R. D.

J. W. Foreman, E. W. George, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flow meter,” Appl. Phys. Lett., vol. 7, p. 77, August5, 1965.
[Crossref]

McMurtry, B. J.

A. E. Siegman, S. E. Harris, B. J. McMurtry, “Optical heterodyning and optical demodulation at microwave frequencies,” in Optical Masers, J. Fox, Ed. New York: Polytechnic Press and Wiley, 1963, p. 511.

Oliver, B. M.

H. A. Haus, C. H. Townes, B. M. Oliver, “Comments on ‘Noise in photoelectric mixing,’” Proc. IRE (Correspondence), vol. 50, pp. 1544–1545, June1962.

B. M. Oliver, “Signal-to-noise ratios in photoelectric mixing,” Proc. IRE (Correspondence), vol. 49, pp. 1960–1961, December1961.

Rabinowitz, P.

P. Rabinowitz, J. LaTourrette, G. Gould, “AFC optical heterodyne detector,” Proc. IEEE (Correspondence), vol. 51, pp. 857–858, May1963.
[Crossref]

S. Jacobs, P. Rabinowitz, “Optical heterodyning with a CW gaseous laser,” in Quantum Electronics III, P. Grivet, N. Bloembergen, Eds. New York: Columbia University Press1964, p. 481.

Read, W. S.

W. S. Read, R. G. Turner, “Tracking heterodyne detection,” Appl. Opt., vol. 4, p. 1570, December1965.
[Crossref]

W. S. Read, D. L. Fried, “Optical heterodyning with noncritical angular alignment,” Proc. IEEE (Correspondence), vol. 51, p. 1787, December1963.
[Crossref]

Siegman, A. E.

A. E. Siegman, S. E. Harris, B. J. McMurtry, “Optical heterodyning and optical demodulation at microwave frequencies,” in Optical Masers, J. Fox, Ed. New York: Polytechnic Press and Wiley, 1963, p. 511.

A. E. Siegman, “Detection and demodulation of laser beams,” presented at the 1964 International Symposium, IEEE Microwave Theory and Techniques Group, New York, N. Y.

A. E. Siegman, “Spatially coherent detection of scattered radiation,” submitted for publication, IEEE Trans. on Antennas and Propagation.

Smith, R. L.

Townes, C. H.

H. A. Haus, C. H. Townes, B. M. Oliver, “Comments on ‘Noise in photoelectric mixing,’” Proc. IRE (Correspondence), vol. 50, pp. 1544–1545, June1962.

Turner, R. G.

Yeh, Y.

H. Z. Cummins, N. Knable, Y. Yeh, “Observation of diffusion broadening of Rayleigh scattered light,” Phys. Rev. Lett., vol. 12, p. 150, February10, 1964.
[Crossref]

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurements with an He-Ne laser spectrometer,” Appl. Phys. Lett., vol. 4, p. 176May15, 1964.
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurements with an He-Ne laser spectrometer,” Appl. Phys. Lett., vol. 4, p. 176May15, 1964.
[Crossref]

J. W. Foreman, E. W. George, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flow meter,” Appl. Phys. Lett., vol. 7, p. 77, August5, 1965.
[Crossref]

Electronics (1)

S. Jacobs, “The optical heterodyne,” Electronics, vol. 36(28), p. 29, July12, 1963.

J. Appl. Phys. (1)

V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys., vol. 36, p. 1819, June1965.
[Crossref]

Phys. Rev. Lett. (1)

H. Z. Cummins, N. Knable, Y. Yeh, “Observation of diffusion broadening of Rayleigh scattered light,” Phys. Rev. Lett., vol. 12, p. 150, February10, 1964.
[Crossref]

Proc. IEEE (Correspondence) (4)

P. Rabinowitz, J. LaTourrette, G. Gould, “AFC optical heterodyne detector,” Proc. IEEE (Correspondence), vol. 51, pp. 857–858, May1963.
[Crossref]

A. J. Bahr, “The effect of polarization selectivity on optical mixing in photoelectric surfaces,” Proc. IEEE (Correspondence), vol. 53, p. 513, May1965.
[Crossref]

R. D. Kroger, “Motion sensing by optical heterodyne Doppler detection from diffuse surfaces,” Proc. IEEE (Correspondence)vol. 53, pp. 211–212, February1965.
[Crossref]

W. S. Read, D. L. Fried, “Optical heterodyning with noncritical angular alignment,” Proc. IEEE (Correspondence), vol. 51, p. 1787, December1963.
[Crossref]

Proc. IRE (Correspondence) (2)

B. M. Oliver, “Signal-to-noise ratios in photoelectric mixing,” Proc. IRE (Correspondence), vol. 49, pp. 1960–1961, December1961.

H. A. Haus, C. H. Townes, B. M. Oliver, “Comments on ‘Noise in photoelectric mixing,’” Proc. IRE (Correspondence), vol. 50, pp. 1544–1545, June1962.

Other (6)

S. Jacobs, P. Rabinowitz, “Optical heterodyning with a CW gaseous laser,” in Quantum Electronics III, P. Grivet, N. Bloembergen, Eds. New York: Columbia University Press1964, p. 481.

A. E. Siegman, S. E. Harris, B. J. McMurtry, “Optical heterodyning and optical demodulation at microwave frequencies,” in Optical Masers, J. Fox, Ed. New York: Polytechnic Press and Wiley, 1963, p. 511.

This same point has been noted by R. E. Brooks, TRW Space Technology Labs, Redondo Beach, Calif., unpublished memorandum.

Note added in proof:More recent calculations indicate that the limit seems to be given by Nσλ/4, i.e., a factor of πlarger than stated in the text. This “theorem” has yet to be proved in general, but an exact calculation for a single transmit-receive antenna with a Gaussian beam pattern leads to exactly Nσλ/4.

A. E. Siegman, “Detection and demodulation of laser beams,” presented at the 1964 International Symposium, IEEE Microwave Theory and Techniques Group, New York, N. Y.

A. E. Siegman, “Spatially coherent detection of scattered radiation,” submitted for publication, IEEE Trans. on Antennas and Propagation.

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Figures (4)

Fig. 1
Fig. 1

General schematic of an optical heterodyne receiver.

Fig. 2
Fig. 2

Several sketches illustrating various concepts in optical heterodyne antenna theory.

Fig. 3
Fig. 3

Geometry for the analysis of heterodyne detection of scattered light.

Fig. 4
Fig. 4

Heterodyne detection of scattered light using the same aperture for transmitting and receiving.

Equations (37)

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d I ( t ) = η ( x , y ) u ˜ ( x , y , t ) 2 d A
I ( t ) = η ( x , y ) u ˜ ( x , y , t ) 2 d A ,
u ˜ ( x , y , t ) = u ˜ 0 ( x , y ) e j ω 0 t + u ˜ 1 ( x , y ) e j ω 1 t .
I ( t ) = I 0 + I 1 + 1 2 [ I ˜ 10 e j ( ω 1 - ω 0 ) t + I ˜ 10 * e - j ( ω 1 - ω 0 ) t ] ,
I 0 = η ( x , y ) u ˜ 0 ( x , y ) 2 d A I 1 = η ( x , y ) u ˜ 1 ( x , y ) 2 d A ,
1 2 I ˜ 10 = η ( x , y ) u ˜ 1 ( x , y ) u ˜ 0 * ( x , y ) d A .
I ˜ 10 2 = 4 I 0 I 1
u ˜ 1 ( x , y , z ) = e ˜ 1 e - j k · r .
u ˜ 1 ( x , y , z = 0 ) = e ˜ 1 e - j ( k x x + k y y ) .
k x = k 1 sin θ cos ϕ k y = k 1 sin θ sin ϕ .
1 2 I ˜ 10 = e ˜ 1 η ( x , y ) u 0 * ( x , y ) e - j ( k x x + k y y ) d x d y .
η ¯ η ( x , y ) u 0 ( x , y ) 2 d x d y u 0 ( x , y ) 2 d x d y
I ˜ 10 2 = 4 I 0 [ η ˜ e ˜ 1 2 A R ] .
A R ( k x , k y ) = A R ( Ω ) = | η ( x , y ) u 0 * ( x , y ) e - j ( k x x + k y y ) d x d y | 2 η ¯ 2 u 0 ( x , y ) 2 d x d y
d Ω = sin θ d θ d ϕ .
d k x d k y = k 1 2 cos θ sin θ d θ d ϕ = k 1 2 cos θ d Ω .
all solid angle A R ( Ω ) d Ω ( 1 / k 1 2 ) - A R ( k x , k y ) d k x d k y .
1 k 1 2 A R ( k x , k y ) d k x d k y = ( 2 π k 1 ) 2 η 2 ( x , y ) u 0 ( x , y ) 2 d x d y η ¯ 2 u 0 ( x , y ) 2 d x d y
A R ( Ω ) d ( Ω ) = η 2 ¯ η ¯ 2 λ 2 ,
A R ( Ω ) d Ω λ 2
A R Ω R λ 2 .
A R Ω R A × ( λ 2 A ) λ 2 ,
Ω R = A C f 2
λ 2 A L A S f 2 .
A R = ( A S A C ) × A L .
A R Ω R ( A C f 2 ) × ( A S A L A C ) = A S A L f 2 λ 2 ,
A R Ω R ( M 2 A R ) × ( 1 M 2 Ω R ) λ 2 .
P received = ( N a b c sin θ ) ( A R 4 π R 2 ) ( σ a b sin θ ) P trans = λ b sin θ N σ λ 4 π P trans .
P received P trans N σ λ 4 π .
d P received = ( N Ω R r 2 d r ) ( A R 4 π r 2 ) ( σ r 2 Ω R ) P trans = N σ A R 4 π d r r 2 P trans .
r 0 A R Ω R A R λ .
P received P trans = N σ A R 4 π r = r 0 d r r 2 = N σ λ 4 π .
Δ P = 2 π λ 2 h f Δ f e h f / k T - 1 Δ A .
P sig = ( 1 2 ) ( A R π R 2 ) ( 2 π λ 2 h f d f e h f / k T - 1 ) ( R 2 Ω R ) = h f d f e h f / k T - 1 .
P noise = ( 1 / η ¯ ) h f d f .
S N P sig P noise = η ¯ e h f / k T - 1 .
T h f k 1 ln ( 1 + η ¯ ) h f k 1 η ¯ ,             η ¯ 1.

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